Weighted Poincaré-type Estimates for Conjugate A-harmonic Tensors
نویسنده
چکیده
In a survey paper [1], Agarwal and Ding summarized the advances achieved in the study of A-harmonic equations. Some recent results about A-harmonic equations can also be found in [2, 3, 5, 6]. The purpose of this note is to establish some estimates about Green’s operator and the Hodge codifferential operator d∗, which will enrich the existing literature in the field of A-harmonic equations. Let Ω be a connected open subset of Rn, n≥ 2, B a ball in Rn and ρB denote the ball with the same center as B and with diam(ρB)= ρdiam(B). The n-dimensional Lebesgue measure of a set E ⊆ Rn is denoted by |E|. We call w a weight if w ∈ Lloc(R) and w > 0 a.e. For 0 < p < ∞ and a weight w(x), we denote the weighted Lp-norm of a measurable function f over E by ‖ f ‖p,E,wα = ( ∫ E | f (x)|pwαdx)1/p, where α is a real number. Let Λl = Λl(Rn) be the linear space of all l-forms ω(x) = ∑I ωI(x)dxI = ∑ωi1i2···il(x)dxi1 ∧ dxi2 ∧ ··· ∧ dxil , l = 0,1, . . . ,n. Assume that D′(Ω,Λl) is the space of all differential lforms and Lp(Ω,Λl) is the space of all Lp-integrable l-forms, which is a Banach space with norm ‖ω‖p,Ω = ( ∫ Ω |ω(x)|pdx)1/p = ( ∫ Ω( ∑ I |ωI(x)|2)p/2dx)1/p. We denote the exterior derivative by d : D′(Ω,Λl) → D′(Ω,Λl+1) for l = 0,1, . . . ,n− 1. Its formal adjoint operator d∗ : D′(Ω,Λl+1)→ D′(Ω,Λl) is given by d∗ = (−1)nl+1∗d∗ on D′(Ω,Λl+1), l = 0,1, . . . ,n− 1, where ∗ is the Hodge star operator. We call u and v a pair of conjugate A-harmonic tensor in Ω if u and v satisfy the conjugate A-harmonic equation
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تاریخ انتشار 2005